Hypothesis Testing 4--Chi-Square for Goodness of Fit Tests and
Independence
In Hypothesis Testing 1, 2 and 3, you have used normal and
t-distributions to test hypotheses. Chi-Square tests use the Chi-Square
probability distribution. You will be introduced to the use of that
distribution in Goodness of Fit tests and tests for Independence.
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Goodness of Fit Tests
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Assumptions
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Example--Die
You are given a die that is said to be fair. You
decide to test this by tossing the die 120 times and testing at the 5%
level of significance.. In those 120 tosses you observe the following
outcomes.
| Side |
1 |
2 |
3 |
4 |
5 |
6 |
| Observed |
20 |
10 |
30 |
20 |
30 |
10 |
The null hypothesis is H0: Die is fair (p1=p2=p3=p4=p5=p6=1/6
where pi is the probability of side i on the die), and the alternative
hypothesis is Ha: Die is not fair (Not (p1=p2=p3=p4=p5=p6=1/6)).
If the die is fair you would expect each side to appear about 20 times
(120*1/6). Then the last table can be augmented as follows:
| Side |
1 |
2 |
3 |
4 |
5 |
6 |
| Observed |
20 |
10 |
30 |
20 |
30 |
10 |
| Expected |
20 |
20 |
20 |
20 |
20 |
20 |
If the null hypothesis is true, the observed and
expected values should be
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Example--Traffic
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Tests for Independence
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